# Quantum Hamiltonian simulation
### Overview
Quantum Hamiltonian simulation is a class of problems targeting at implementing the map$$H \mapsto \exp(-i \tau H)$$for some Hermitian matrix $$H$$. Using QSP, the quantum hamiltonian simulation boils down to setting the target scalar function as $$f(x) = \exp(-i\tau x)$$. Due to the parity constraint, the implementation of the target scalar function is separated into implementing components $$f\_\mathrm{Re}(x) = \cos(\tau x)$$ and $$f\_\mathrm{Im}(x) = \sin(\tau x)$$ respectively. Then, these components are combined to realize $$\exp(-i\tau H)$$ using linear combination of unitaries (LCU). To illustrate the workflow, we set the evolution time to $$\tau = 100$$.
```matlab
tau = 100
% set the parameters of the solver
opts.maxiter = 100;
opts.criteria = 1e-12;
% use the real representation to speed up the computation
opts.useReal = true;
% set the solver to fixed-point iteration
opts.method = 'CM';
```
### Solving for the real component
To increase the numerical stability, the (real component of the) target function is scaled down by a factor $$1/2$$. Then, the target function is uniformly upper bounded by $$1/2$$. The Chebyshev coefficients of the target function is obtained by truncating the series to some finite degree. According to the analysis, it suffices to set $$d = 1.4 |\tau| + \log\left(\frac{1}{\epsilon\_0}\right)$$so that the truncation error is lower than $$\epsilon\_0$$.
{% code fullWidth="false" %}
```matlab
targ = @(x) 0.5*cos(tau.*x);
parity = 0; % indicating the even parity
% compute the Chebyshev coefficients
d = ceil(1.4*tau+log(1e14));
f = chebfun(targ,d);
coef = chebcoeffs(f);
% discard coefficients of odd orders due to the even parity
coef = coef(parity+1:2:end);
```
{% endcode %}
Start the solver to find phase factors.
```matlab
[phi_proc,out] = QSP_solver(coef,parity,opts);
```
### Verifying the solution
We verify the solved phase factors by computing the residual error in terms of the normalized $$\ell^1$$ norm
$$
\mathrm{residual\_error} = \frac{1}{K} \sum\_{k=1}^K |g(x\_k,\Phi^\*) - f(x\_k)|.
$$
Using $$1,000$$ equally spaced points, the residual error is $$9.4988 \times 10^{-14}$$ which attains almost machine precision. We also plot the pointwise error.
```matlab
xlist = linspace(0, 1, 1000)';
targ_value = targ(xlist);
QSP_value = QSPGetEntry(xlist, phi_proc, out);
err= norm(QSP_value-targ_value,1)/length(xlist);
disp('The residual error is');
disp(err);
% plot the pointwise error
plot(xlist,QSP_value-targ_value)
xlabel('$$x$$', 'Interpreter', 'latex')
ylabel('$$g(x,\Phi^*)-f(x)$$', 'Interpreter', 'latex')
```
The point-wise error of the solved phase factors.
### Reference
1. Low, G. H., & Chuang, I. L. (2017). Optimal Hamiltonian simulation by quantum signal processing. *Physical review letters*, *118*(1), 010501.
2. Gilyén, A., Su, Y., Low, G. H., & Wiebe, N. (2019, June). Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In *Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing* (pp. 193-204).
3. Dong, Y., Meng, X., Whaley, K. B., & Lin, L. (2021). Efficient phase-factor evaluation in quantum signal processing. *Physical Review A*, *103*(4), 042419.
Output of the code
```
iter err
1 +1.7869e-01
2 +2.6363e-02
3 +3.4033e-03
4 +3.8478e-04
5 +4.0011e-05
6 +4.7448e-06
7 +6.9274e-07
8 +1.0545e-07
9 +1.5579e-08
10 +2.2009e-09
iter err
11 +3.1451e-10
12 +4.3592e-11
13 +6.0816e-12
14 +8.6952e-13
Stop criteria satisfied.
The residual error is
9.4988e-14
```
---
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